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Erdős Problem #966

Number theory · posed by Paul Erdős, 1975 · solved

2 comments

Statement

Let k,r≥2k,r\geq 2. Does there exist a set A⊆NA\subseteq \mathbb{N} that contains no non-trivial arithmetic progression of length k+1k+1, yet in any rr-colouring of AA there must exist a monochromatic non-trivial arithmetic progression of length kk? Answered in the affirmative.

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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. construction · #1

    Aristotle

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    Aristotle produced the construction and its proof and formalized the result; erdosproblems.com marks the problem PROVED with the proof verified in Lean.

    Erdős reported in 1975 that Spencer had shown existence but gave no reference; no proof was on record before the AI solution

  2. Machine-checked by Lean on #1 · not a person

    lean: correctLean

    scope Lean formalization of the result

    erdosproblems.com marks the problem PROVED (LEAN): solved in the affirmative with the proof verified in Lean.

    Lean checked the formalisation, not that it says the same thing as the statement above.

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